1

The wavelike properties of particlesThe wavelike properties of particles

Cat: “Am I a particle or wave?”Cat: “Am I a particle or wave?”

2

Wave particle dualityWave particle duality

•• ““Quantum nature of light” refers to the Quantum nature of light” refers to the particle attribute of lightparticle attribute of light

•• “Quantum nature of particle” refers to the “Quantum nature of particle” refers to the wave attribute of a particlewave attribute of a particle

•• Light (classically EM waves) is said to Light (classically EM waves) is said to display “wavedisplay “wave--particle duality” particle duality” –– it behave it behave like wave in one experiment but as particle like wave in one experiment but as particle in others (c.f. a person with schizophrenia)in others (c.f. a person with schizophrenia)

3

•• Not only light does have “schizophrenia”, so are Not only light does have “schizophrenia”, so are other microscopic ``particle’’ such as electron, other microscopic ``particle’’ such as electron, (see later chapters), i.e. particle” also manifest (see later chapters), i.e. particle” also manifest wave characteristics in some experimentswave characteristics in some experiments

•• WaveWave--particle duality is essentially the particle duality is essentially the manifestation of the quantum nature of thingsmanifestation of the quantum nature of things

•• This is an very weird picture quite contradicts to This is an very weird picture quite contradicts to our conventional assumption with is deeply rooted our conventional assumption with is deeply rooted on classical physics or intuitive notion on thingson classical physics or intuitive notion on things

4

Planck constant as a measure of Planck constant as a measure of quantum effectquantum effect

•• When investigating physical systems involving its When investigating physical systems involving its quantum nature, the theory usually involves the quantum nature, the theory usually involves the appearance of the constant appearance of the constant hh

•• i.e. in Compton scattering, the Compton shift is i.e. in Compton scattering, the Compton shift is proportional to proportional to hh; So is ; So is photoelectricityphotoelectricity involves involves hh in its in its formulaformula

•• In general, when In general, when hh appears, it means quantum effects appears, it means quantum effects arisearise

•• In contrary, in classical mechanics or classical EM In contrary, in classical mechanics or classical EM theory, theory, hh never appear both theories do not take into never appear both theories do not take into account quantum effects, hence the constant account quantum effects, hence the constant hh never never appear in such theoriesappear in such theories

•• Roughly quantum effects arise in microscopic system Roughly quantum effects arise in microscopic system (e.g. on the scale approximately of the order 10(e.g. on the scale approximately of the order 10--1010 m or m or smaller)smaller)

5

Wavelike properties of particleWavelike properties of particle•• In 1923, while still a graduate In 1923, while still a graduate

student at the University of student at the University of Paris, Louis de Paris, Louis de BroglieBrogliepublished a brief note in the published a brief note in the journal journal ComptesComptes rendusrenduscontaining an idea that was to containing an idea that was to revolutionize our understanding revolutionize our understanding of the physical world at the of the physical world at the most fundamental level: most fundamental level:

•• That particle has intrinsic wave That particle has intrinsic wave propertiesproperties

•• For more interesting details:For more interesting details:•• http://www.davishttp://www.davis--

inc.com/physics/index.shtmlinc.com/physics/index.shtmlPrince de Broglie, 1892-1987

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de de Broglie’sBroglie’s postulate (1924)postulate (1924)

•• The postulate: there should be a symmetry The postulate: there should be a symmetry between matter and wave. The wave aspect of between matter and wave. The wave aspect of matter is related to its particle aspect in exactly the matter is related to its particle aspect in exactly the same quantitative manner that is in the case for same quantitative manner that is in the case for radiation. The total (i.e. relativistic) energy radiation. The total (i.e. relativistic) energy EE and and momentum momentum p p of an entity, for both matter and of an entity, for both matter and wave alike, is related to the frequency wave alike, is related to the frequency νν of the of the wave associated with its motion via by Planck wave associated with its motion via by Planck constantconstant

E = E = hfhf, p , p = = hh//λλ

7

A particle has wavelength!!!A particle has wavelength!!!

λλ = = hh//pp•• is the de is the de BroglieBroglie relation predicting the wave length of the matter relation predicting the wave length of the matter

wave wave λλ associated with the motion of a material particle with associated with the motion of a material particle with momentum momentum pp

•• Note that classically the property of wavelength is only reserveNote that classically the property of wavelength is only reserved for d for wave and particle was never associate with any wavelengthwave and particle was never associate with any wavelength

•• But, following de But, following de BroglieBroglie’’ss postulate, such distinction is removedpostulate, such distinction is removed

Particle with linear momentum p

Matter wave with de Broglie wavelength

λ = p/h

A particle with momentum pis pictured as a wave (wavepulse)

8

A physical entity possess both aspects A physical entity possess both aspects of particle and wave in a of particle and wave in a complimentary mannercomplimentary manner

BUT why is the wave nature of material particle

not observed?

Because …

9

•• BecauseBecause……we are too large and quantum effects are too smallwe are too large and quantum effects are too small

•• Consider two extreme cases: Consider two extreme cases: •• (i) an electron with kinetic energy (i) an electron with kinetic energy KK = 54 = 54 eVeV, de , de BroglieBroglie wavelength, wavelength, λλ

= = hh//pp = = hh / (2/ (2mmeeKK))1/21/2 = 1.65 Angstrom. = 1.65 Angstrom. •• Such a wavelength is comparable to the size of atomic lattice, aSuch a wavelength is comparable to the size of atomic lattice, and is nd is

experimentally detectableexperimentally detectable

•• (ii) As a comparison, consider an macroscopic object, a (ii) As a comparison, consider an macroscopic object, a billardbillard ball of ball of mass mass mm = 100 g moving with momentum = 100 g moving with momentum pp

•• pp = = mvmv ≈≈ 0.1 kg 0.1 kg ×× 10 10 m/sm/s = 1 Ns (relativistic correction is negligible)= 1 Ns (relativistic correction is negligible)•• It has de It has de BroglieBroglie wavelength wavelength λλ = = hh//pp ≈≈ 1010--3434 m, too tiny to be observed m, too tiny to be observed

in any experimentsin any experiments•• The total energy of the The total energy of the billardbillard ball is ball is

•• E = K + mE = K + m00cc2 2 == mcmc2 2 = 0.1= 0.1×× (3 (3 ××101088))22 J = 9 J = 9 ×× 10101515J J •• The frequency of the de The frequency of the de BroglieBroglie wave associated with the wave associated with the billardbillard ball is ball is ff = = EE//hh = = mcmc22//hh= (9= (9××10101515/6.63/6.63××10103434) Hz = (9) Hz = (9××10101515/6.63/6.63××10103434) =10) =1078 78

Hz, impossibly high for any experiment to detectHz, impossibly high for any experiment to detect10

Matter wave is a quantum Matter wave is a quantum phenomenaphenomena

•• This also means that the wave properties of matter is This also means that the wave properties of matter is difficult to observe for macroscopic system (unless with difficult to observe for macroscopic system (unless with the aid of some specially designed apparatus)the aid of some specially designed apparatus)

•• The smallness of The smallness of hh in the relation in the relation λλ = = hh//pp makes wave makes wave characteristic of particles hard to be observedcharacteristic of particles hard to be observed

•• The statement that when The statement that when h h 00, , λ λ becomes vanishingly becomes vanishingly small means that: small means that:

•• the wave nature will become effectively “shutthe wave nature will become effectively “shut--off” and off” and appear to loss its wave nature whenever the relevant appear to loss its wave nature whenever the relevant pp of of the particle is too large in comparison with the quantum the particle is too large in comparison with the quantum scale characterised by scale characterised by hh

•• More quantitatively, we could not detect the quantum More quantitatively, we could not detect the quantum effect if effect if h/ph/p ~ 10~ 10--3434 Js/Js/pp becomes too tiny in comparison becomes too tiny in comparison to the length scale discernable by an experimental setupto the length scale discernable by an experimental setup

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The The thethe particleparticle’’s velocity s velocity vv00 is is identified with the de identified with the de BroglieBroglie’’ group group

wave, wave, vvgg but not its phase wave but not its phase wave vvpp

vvpp,, could be could be larger than larger than cc

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ExampleExample• An electron has a de Broglie wavelength of 2.00

pm. Find its kinetic energy and the phase and the group velocity of its de Broglie waves.

• Total energy E2 = c2p2 + m0i2

• K = E - m0c2 = (c2p2 + m02c4) ½- m0c2 =

• ((hc/λ)2 + m02c4) ½- m0c2= 297 keV

• vg= v; 1/γ2 = 1 – (v/c)2;• (pc)2 = (γm0vc)2= (hc/λ)2 (from Relativity and de

Broglie’s postulate)⇒(γv/c)2= (hc/λ)2/(m0c2)2=(620 keV/510 keV)2 =

1.4884;(γv/c)2 = (v/c)2 / 1- (v/c)2

⇒ vg /c=√(1.4884/(1+1.4884))=0.77

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Interference experiment with a Interference experiment with a single electron, firing one in a timesingle electron, firing one in a time

•• Consider an double slit Consider an double slit experiment using an experiment using an extremely electron source extremely electron source that emits only one that emits only one electron a time through electron a time through the double slit and then the double slit and then detected on a fluorescent detected on a fluorescent plateplate

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•• When one follows the time evolution of the pattern When one follows the time evolution of the pattern created by these individual photons, what sort of created by these individual photons, what sort of pattern do you think you will observed?pattern do you think you will observed?

•• It’s the interference pattern that are in fact observed It’s the interference pattern that are in fact observed in experimentsin experiments

•• At the source the electron is being emitted as particle At the source the electron is being emitted as particle and is experimentally detected as a electron which is and is experimentally detected as a electron which is absorbed by an individual atom in the fluorescent absorbed by an individual atom in the fluorescent plateplate

•• In between, we must interpret the electron in the form In between, we must interpret the electron in the form of a wave. The double slits change the propagation of of a wave. The double slits change the propagation of the electron wave so that it is ‘processed’ to forms the electron wave so that it is ‘processed’ to forms diffraction pattern on the screen. diffraction pattern on the screen.

•• Such process would be impossible if electrons are Such process would be impossible if electrons are particle (because no one particle can go through both particle (because no one particle can go through both slits at the same time. Such a simultaneous slits at the same time. Such a simultaneous penetration is only possible for wave.)penetration is only possible for wave.)

•• Be reminded that the wave nature in the intermediate Be reminded that the wave nature in the intermediate states is not measured. Only the particle nature are states is not measured. Only the particle nature are detected in this procedure.detected in this procedure.

Electrons display interference Electrons display interference patternpattern

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Detection of electron as particle Detection of electron as particle destroy the interference patterndestroy the interference pattern

•• If in the electron interference If in the electron interference experiment one tries to place a experiment one tries to place a detector on each hole to determine detector on each hole to determine though which an electron passes, the though which an electron passes, the wave nature of electron in the wave nature of electron in the intermediate states are destroyed intermediate states are destroyed

•• e.g. by the interference pattern on e.g. by the interference pattern on the screen shall be destroyedthe screen shall be destroyed

•• Why? It is the consistency of the Why? It is the consistency of the wavewave--particle duality that demands particle duality that demands such destruction must happen (think such destruction must happen (think of the logics yourself or read up of the logics yourself or read up from the text)from the text)

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•• The correct explanation of the origin and The correct explanation of the origin and appearance of the interference pattern comes from appearance of the interference pattern comes from the wave picture, and the correct interpretation of the wave picture, and the correct interpretation of the evolution of the pattern on the screen comes the evolution of the pattern on the screen comes from the particle picture;from the particle picture;

•• Hence to completely explain the experiment, the Hence to completely explain the experiment, the two pictures must somehow be taken together two pictures must somehow be taken together ––this is an example for which this is an example for which both pictures are both pictures are complimentary to each othercomplimentary to each other

•• Try to compare the last few slides with slide 119 Try to compare the last few slides with slide 119 from previous chapter for photon, which also from previous chapter for photon, which also displays wavedisplays wave--particle dualityparticle duality

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So, is electron wave or particleSo, is electron wave or particle??•• They are bothThey are both……but not but not

simultaneouslysimultaneously•• In any experiment (or empirical In any experiment (or empirical

observation) only one aspect of observation) only one aspect of either wave or particle, but not either wave or particle, but not both can be observed both can be observed simultaneously. simultaneously.

•• ItIt’’s like a coin with two faces. s like a coin with two faces. But one can only see one side But one can only see one side of the coin but not the other at of the coin but not the other at any instanceany instance

•• This is the soThis is the so--called wavecalled wave--particle dualityparticle duality

Electron as particle

Electron as wave

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“Once and for all I want to know what I’m paying for. When the electric company tells me whether electron is a wave or a particle I’ll write my check”

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Extra readingsExtra readings

•• Those quantum enthusiasts may like to read more Those quantum enthusiasts may like to read more about waveabout wave--particle duality in Section 5.7, page particle duality in Section 5.7, page 179179--185, 185, SerwaySerway and Mosesand Moses

•• An even more recommended reading on waveAn even more recommended reading on wave--particle duality: the Feynman lectures on physics, particle duality: the Feynman lectures on physics, vol. III, chapter 1 (Addisonvol. III, chapter 1 (Addison--Wesley Publishing)Wesley Publishing)

•• ItIt’’s a very interesting and highly intellectual topic s a very interesting and highly intellectual topic to investigateto investigate

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DavissonDavisson and and GremerGremer experimentexperiment

•• DG confirms the wave DG confirms the wave nature of electron in nature of electron in which it undergoes which it undergoes BraggBragg’’s diffractions diffraction

•• ThermionicThermionic electrons are electrons are produced produced by hot filament, by hot filament, accelerated and focused accelerated and focused onto the target (all onto the target (all apparatus is in vacuum apparatus is in vacuum condition)condition)

•• Electrons are scattered at Electrons are scattered at an angle an angle φφ into a movable into a movable detectordetector

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Pix of Davisson and Gremer

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Result of the DG experimentResult of the DG experiment

•• Distribution of electrons Distribution of electrons is measured as a is measured as a function of function of φφ

•• Strong scattered eStrong scattered e--beam is detected atbeam is detected at φφ = = 50 degree for50 degree for V = V = 5454 VV

23

How to interpret the result of DG?How to interpret the result of DG?

•• Electrons get diffracted by Electrons get diffracted by the atoms on the surface the atoms on the surface (which acted as diffraction (which acted as diffraction grating) of the metal as grating) of the metal as though the electron acting though the electron acting like they are WAVE like they are WAVE

•• Electron do behave like Electron do behave like wave as postulated by de wave as postulated by de BroglieBroglie

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Constructive BraggConstructive Bragg’’s diffractions diffraction

•• The peak of the diffraction pattern is the The peak of the diffraction pattern is the m=1m=1stst order constructive interference: order constructive interference: ddsinsin φ φ == 11λλ

•• where where φφ = = 50 degree for50 degree for V = V = 5454 VV•• From xFrom x--ray Bragg’s diffraction ray Bragg’s diffraction

experiment done independentlyexperiment done independently we know we know d d = 2.15 = 2.15 AmstrongAmstrong

•• Hence the wavelength of the electron is Hence the wavelength of the electron is λλ= = ddsinsinθθ = = 1.65 Angstrom 1.65 Angstrom

•• Here, 1.65 Here, 1.65 Angstrom is the Angstrom is the experimentally inferred value, which is to experimentally inferred value, which is to be checked against the theoretical value be checked against the theoretical value predicted by de predicted by de BroglieBroglie

φ

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Theoretical value of Theoretical value of λλ of the electronof the electron

•• An external potential An external potential V V accelerates the electron via accelerates the electron via eVeV==K K

•• In the DG experiment the kinetic energy of the In the DG experiment the kinetic energy of the electron is accelerated to electron is accelerated to KK = 54 = 54 eVeV (non(non--relativistic relativistic treatment is suffice because treatment is suffice because KK << << mmeecc22 = 0.51 = 0.51 MeVMeV) )

•• According to de According to de BroglieBroglie, the wavelength of an electron , the wavelength of an electron accelerated to kinetic energy of accelerated to kinetic energy of KK = = pp22/2/2mmee = 54 = 54 eVeVhas a equivalent matter wave wavelength has a equivalent matter wave wavelength λλ = = hh//pp = = hh/(2/(2KmKmee))--1/2 1/2 = 1.67 = 1.67 AmstrongAmstrong

•• In terms of the external potential, In terms of the external potential, λλ = = hh/(2/(2eVmeVmee))--1/21/2

26

Theory’s prediction matches Theory’s prediction matches measured valuemeasured value

•• The result of DG measurement agrees almost The result of DG measurement agrees almost perfectly with the de perfectly with the de BroglieBroglie’’ss predictionprediction: : 1.65 1.65 Angstrom measured by DG experiment against Angstrom measured by DG experiment against 1.67 Angstrom according to theoretical prediction1.67 Angstrom according to theoretical prediction

•• Wave nature of electron is hence experimentally Wave nature of electron is hence experimentally confirmedconfirmed

•• In fact, wave nature of microscopic particles are In fact, wave nature of microscopic particles are observed not only in eobserved not only in e-- but also in other particles but also in other particles (e.g. neutron, proton, molecules etc. (e.g. neutron, proton, molecules etc. –– most most strikingly Bosestrikingly Bose--Einstein condensateEinstein condensate))

27

Application of electrons wave:Application of electrons wave:electron microscope, Nobel Prize electron microscope, Nobel Prize

1986 (Ernst 1986 (Ernst RuskaRuska))

28

•• ElectronElectron’’s de s de BroglieBrogliewavelength can be tuned wavelength can be tuned via via λλ = = hh/(2/(2eVmeVmee))--1/21/2

•• Hence electron Hence electron microscope can magnify microscope can magnify specimen (x4000 times) specimen (x4000 times) for biological specimen for biological specimen or 120,000 times of wire or 120,000 times of wire of about 10 atoms in of about 10 atoms in widthwidth

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Not only electron, other microscopicNot only electron, other microscopicparticles also behave like wave at the particles also behave like wave at the

quantum scalequantum scale•• The following atomic structural images provide insight into the The following atomic structural images provide insight into the threshold threshold

between prime radiant flow and the interference structures callebetween prime radiant flow and the interference structures called d matter.matter.

•• In the right foci of the ellipse a real cobalt atom has been insIn the right foci of the ellipse a real cobalt atom has been inserted. In the erted. In the left foci of the ellipse a phantom of the real atom has appearedleft foci of the ellipse a phantom of the real atom has appeared. The . The appearance of the phantom atom was not expected.appearance of the phantom atom was not expected.

•• The ellipsoid coral was constructed by placing 36 cobalt atom onThe ellipsoid coral was constructed by placing 36 cobalt atom on a a copper surface. This image is provided here to provide a visual copper surface. This image is provided here to provide a visual demonstration of the attributes of material matter arising from demonstration of the attributes of material matter arising from the the harmonious interference of background radiation.harmonious interference of background radiation.

QUANTUM CORAL http://home.netcom.co

m/~sbyers11/grav11E.htm

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Heisenberg’s uncertainty principle Heisenberg’s uncertainty principle (Nobel Prize,1932)(Nobel Prize,1932)

•• WERNER HEISENBERG (1901 WERNER HEISENBERG (1901 -- 1976) 1976) •• was one of the greatest physicists of the was one of the greatest physicists of the

twentieth century. He is best known as a twentieth century. He is best known as a founder of founder of quantum mechanicsquantum mechanics, the new , the new physics of the atomic world, and especially physics of the atomic world, and especially for the for the uncertainty principleuncertainty principle in quantum in quantum theory. He is also known for his theory. He is also known for his controversial role as a leader of Germany's controversial role as a leader of Germany's nuclear fissionnuclear fission research during World War research during World War II. After the war he was active in II. After the war he was active in elementary particle physics and West elementary particle physics and West German German science policyscience policy. .

•• http://www.aip.org/history/heisenberg/p01.http://www.aip.org/history/heisenberg/p01.htmhtm

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A particle is represented by a wave A particle is represented by a wave packet/pulsepacket/pulse

•• Since we experimentally confirmed that particles Since we experimentally confirmed that particles are wave in nature at the quantum scale are wave in nature at the quantum scale hh (matter (matter wave) we now have to describe particles in term of wave) we now have to describe particles in term of waves (relevant only at the quantum scale)waves (relevant only at the quantum scale)

•• Since a real particle is localised in space (not Since a real particle is localised in space (not extending over an infinite extent in space), the wave extending over an infinite extent in space), the wave representation of a particle has to be in the form of representation of a particle has to be in the form of wave packet/wave pulsewave packet/wave pulse

32

•• As mentioned before, As mentioned before, wavepulsewavepulse/wave packet is /wave packet is formed by adding many waves of different formed by adding many waves of different amplitudes and with the wave numbers amplitudes and with the wave numbers spanning a range of spanning a range of ∆∆kk (or equivalently, (or equivalently, ∆∆λλ))

∆xRecall that k = 2Recall that k = 2ππ//λλ, hence, hence

∆∆k/kk/k = ∆λ= ∆λ//λλ

33

Still remember the Still remember the uncertainty uncertainty relationships for classical waves?relationships for classical waves?

•• As discussed earlier, due to its nature, a wave packet must obeyAs discussed earlier, due to its nature, a wave packet must obey the the uncertainty relationships for classical waves (which are deriveduncertainty relationships for classical waves (which are derivedmathematically with some approximations)mathematically with some approximations)

πλλ 2~

2

~>∆∆≡>∆∆ xkx 1≥∆∆ νt

•• However a more rigorous mathematical treatment (without the However a more rigorous mathematical treatment (without the approximation) gives the exact relationsapproximation) gives the exact relations

2/14

2

≥∆∆≡≥∆∆ xkxπ

λλ πν

41≥∆∆ t

•• To describe a particle with wave packet that is localised over aTo describe a particle with wave packet that is localised over a small small region region ∆∆xx requires a large range of wave number; that is, requires a large range of wave number; that is, ∆∆kk is large. is large. Conversely, a small range of wave number cannot produce a wave Conversely, a small range of wave number cannot produce a wave packet localised within a small distance.packet localised within a small distance. 34

•• A narrow wave packet (small A narrow wave packet (small ∆∆xx) corresponds to a ) corresponds to a large spread of wavelengths (large large spread of wavelengths (large ∆∆kk). ).

•• A wide wave packet (large A wide wave packet (large ∆∆xx) corresponds to a ) corresponds to a small spread of wavelengths (small small spread of wavelengths (small ∆∆kk).).

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Matter wave representing a particleMatter wave representing a particlemust also obey similar wave must also obey similar wave

uncertainty relation uncertainty relation •• For matter waves, for which their momentum For matter waves, for which their momentum

and wavelength are related by and wavelength are related by pp = = hh//λλ, the , the uncertainty relationship of the classical wave uncertainty relationship of the classical wave

•• is translated into is translated into

2≥∆∆ xpx

•• where where •• Prove this relation yourselves (hint: from Prove this relation yourselves (hint: from pp = =

hh//λλ, , ∆∆pp//pp = = ∆∆λλ//λλ))

π2/h=

2/14

2

≥∆∆≡≥∆∆ xkxπ

λλ

36

TimeTime--energy uncertaintyenergy uncertainty

•• Just as implies positionJust as implies position--momentum momentum uncertainty = relation, the classical wave uncertainty = relation, the classical wave uncertainty relation also implies a uncertainty relation also implies a corresponding relation of time and energycorresponding relation of time and energy

2≥∆∆ xpx

πν

41≥∆∆ t

2≥∆∆ tE

;4 2

,2

hh t

E h E h h t E t

νπ

ν ν ν

∆ ∆ ≥ =

= ∆ = ∆ ⇒ ∆ ∆ = ∆ ∆ =∵

•• This uncertainty relation can be easily obtained:This uncertainty relation can be easily obtained:

37

Heisenberg uncertainty relationsHeisenberg uncertainty relations

2≥∆∆ xpx 2

≥∆∆ tE

•• The product of the The product of the uncertainty in uncertainty in momentum momentum (energy) and in (energy) and in position (time) is position (time) is at least as large as at least as large as PlanckPlanck’’s constants constant

38

What meansWhat means

•• It sets the intrinsic lowest possible limits on It sets the intrinsic lowest possible limits on the uncertainties in knowing the values of the uncertainties in knowing the values of ppxxand and xx, no matter how good an experiments , no matter how good an experiments is madeis made

•• It is impossible to specify simultaneously It is impossible to specify simultaneously and with infinite precision the linear and with infinite precision the linear momentum and the corresponding position momentum and the corresponding position of a particleof a particle

2≥∆∆ xpx

39

/2

/2

/2

It is impossible for the product It is impossible for the product ∆∆xx∆∆ppxx to be less than to be less than hh/4/4ππ

40

What meansWhat means•• Uncertainty principle for energy. Uncertainty principle for energy. •• The energy of a system also has inherent uncertainty, The energy of a system also has inherent uncertainty,

∆∆EE•• ∆∆E is E is dependent on the dependent on the time intervaltime interval ∆∆tt during which during which

the system remains in the given states. the system remains in the given states. •• If a system is known to exist in a state of energy If a system is known to exist in a state of energy EE over over

a limited period a limited period ∆∆tt, then this energy is uncertain by at , then this energy is uncertain by at least an amount least an amount hh/(4/(4π∆π∆tt). This corresponds to the ). This corresponds to the ‘‘spreadspread’’ in energy of that state (see next page)in energy of that state (see next page)

•• Therefore, the energy of an object or system can be Therefore, the energy of an object or system can be measured with infinite precision (measured with infinite precision (∆∆E=E=0) only if the 0) only if the object of system exists for an infinite time (object of system exists for an infinite time (∆∆tt→∞→∞))

2≥∆∆ tE

41

What meansWhat means•• A system that remains in a A system that remains in a

metastablemetastable state for a very state for a very long time (large long time (large ∆∆tt) can ) can have a very wellhave a very well--defined defined energy (small energy (small ∆∆EE), but if ), but if remain in a state for only a remain in a state for only a short time (small short time (small ∆∆tt), the ), the uncertainty in energy must uncertainty in energy must be correspondingly greater be correspondingly greater (large (large ∆∆EE). ).

2≥∆∆ tE

42

Conjugate variables Conjugate variables (Conjugate observables)(Conjugate observables)

•• {{ppxx,,xx}, {}, {EE,,tt} are called } are called conjugate conjugate variablesvariables

•• The conjugate variables cannot in The conjugate variables cannot in principle be measured (or known) to principle be measured (or known) to infinite precision simultaneouslyinfinite precision simultaneously

43

Heisenberg’s Heisenberg’s GedankenGedankenexperimentexperiment

•• The U.P. can also be understood from the following The U.P. can also be understood from the following gedankengedanken experiment that experiment that tries to measure the position and momentum of an object, say, antries to measure the position and momentum of an object, say, an electron at a electron at a certain momentcertain moment

•• In order to measure the momentum and position of an electron it In order to measure the momentum and position of an electron it is necessary to is necessary to “interfere” it with some “probe” that will then carries the requ“interfere” it with some “probe” that will then carries the required information ired information back to us back to us –– such as shining it with a photon of say a wavelength of such as shining it with a photon of say a wavelength of λλ

44

•• Let’s say the “unperturbed” electron was Let’s say the “unperturbed” electron was initially located at a “definite” location initially located at a “definite” location x x and with a “definite” momentum and with a “definite” momentum pp

•• When the photon ‘probes’ the electron it When the photon ‘probes’ the electron it will be bounced off, associated with a will be bounced off, associated with a changed in its momentum by some changed in its momentum by some uncertain amount, uncertain amount, ∆∆pp. .

•• ∆∆pp cannot be predicted but must be of the cannot be predicted but must be of the similar order of magnitude as the photon’s similar order of magnitude as the photon’s momentum momentum hh//λλ

•• Hence Hence ∆∆pp ≈≈ hh//λλ•• The longer The longer λ (λ (i.e. less energetici.e. less energetic) ) the smaller the smaller

the uncertainty in the measurement of the the uncertainty in the measurement of the electron’s momentum electron’s momentum

•• In other words, electron cannot be observed In other words, electron cannot be observed without changing its momentumwithout changing its momentum

Heisenberg’s Heisenberg’s GedankenGedankenexperimentexperiment

45

•• How much is the uncertainty in the position How much is the uncertainty in the position of the electron? of the electron?

•• By using a photon of wavelength By using a photon of wavelength λ λ we we cannot determine the location of the cannot determine the location of the electron better than an accuracy of electron better than an accuracy of ∆∆xx = = λλ

•• Hence Hence ∆∆xx ≥≥ λλ•• Such is a fundamental constraint coming Such is a fundamental constraint coming

from optics (from optics (Rayleigh’sRayleigh’s criteria).criteria).•• The shorter the wavelength The shorter the wavelength λ (λ (i.e. more i.e. more

energeticenergetic) ) the smaller the uncertainty in the the smaller the uncertainty in the electron’s positionelectron’s position

Heisenberg’s Heisenberg’s GedankenGedankenexperimentexperiment

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•• However, if shorter wavelength is employed However, if shorter wavelength is employed (so that the accuracy in position is (so that the accuracy in position is increased), there will be a corresponding increased), there will be a corresponding decrease in the accuracy of the momentum, decrease in the accuracy of the momentum, measurement (recall measurement (recall ∆∆pp ≈≈ hh//λλ))

•• A higher photon momentum will disturb the A higher photon momentum will disturb the electron’s motion to a greater extentelectron’s motion to a greater extent

•• Hence there is a ‘zero sum game’ hereHence there is a ‘zero sum game’ here•• Combining the expression for Combining the expression for ∆∆xx and and ∆∆pp, ,

we then have we then have ∆∆pp ∆λ∆λ ≥≥ hh, a result consistent , a result consistent with with ∆∆pp ∆λ∆λ ≥≥ hh/2/2

Heisenberg’s Heisenberg’s GedankenGedankenexperimentexperiment

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Heisenberg’s kioskHeisenberg’s kiosk

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•• A typical atomic nucleus is about 5.0A typical atomic nucleus is about 5.0××1010--1515 m m in radius. Use the uncertainty principle to in radius. Use the uncertainty principle to place a lower limit on the energy an electron place a lower limit on the energy an electron must have if it is to be part of a nucleus must have if it is to be part of a nucleus

ExampleExample

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SolutionSolution•• Letting Letting ∆∆xx = = 5.05.0××1010--1515 m, we have m, we have •• ∆∆pp≥≥hh/(4/(4ππ∆∆xx)=…=1)=…=1.1.1××1010--2020 kgkg⋅⋅m/sm/s

If this is the uncertainty in a nuclear electron’s momentum, If this is the uncertainty in a nuclear electron’s momentum, the momentum the momentum pp must be at lest comparable in magnitude. must be at lest comparable in magnitude. An electron of such a momentum has a An electron of such a momentum has a

•• KE = KE = pc pc ≥≥ 33.3.3××1010--1212 J J = 20.6 = 20.6 MeVMeV >> >> mmeecc22= 0.5 = 0.5 MeVMeV

•• i.e., if electrons were contained within the nucleus, they i.e., if electrons were contained within the nucleus, they must have an energy of at least must have an energy of at least 20.6 20.6 MeVMeV

•• However such an high energy electron from radioactive However such an high energy electron from radioactive nuclei never observed nuclei never observed

•• Hence, by virtue of the uncertainty principle, we conclude Hence, by virtue of the uncertainty principle, we conclude that electrons emitted from an unstable nucleus cannot that electrons emitted from an unstable nucleus cannot comes from within the nucleuscomes from within the nucleus

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ExampleExample

•• A measurement established the position of a A measurement established the position of a proton with an accuracy of proton with an accuracy of ±±11.00.00××1010--1111 m. m. Find the uncertainty in the protonFind the uncertainty in the proton’’s position s position 1.00 s later. Assume 1.00 s later. Assume vv << << cc. .

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SolutionSolution•• Let us call the uncertainty in the Let us call the uncertainty in the protonproton’’s position s position ∆∆xx00 at the time at the time tt = 0. = 0.

•• The uncertainty in its momentum at The uncertainty in its momentum at tt = 0 = 0 is is

∆∆pp≥≥hh/(4/(4π π ∆∆xx00))•• Since Since vv << << cc, the momentum uncertainty is , the momentum uncertainty is

∆∆pp= = mm∆∆vv•• The uncertainty in the protonThe uncertainty in the proton’’s velocity s velocity is is

∆∆vv= = ∆∆pp//mm≥≥ hh/(4/(4π π ∆∆xx00))•• The distance The distance xx of the proton covers in of the proton covers in the time the time tt cannot be known more cannot be known more accurately than accurately than

∆∆xx==tt∆∆vv≥≥ htht/(4/(4π π ∆∆xx00))•• The value of at The value of at tt = 1.00 s is 3.15 = 1.00 s is 3.15 kkm. m.

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A moving wave packet spreads out A moving wave packet spreads out in spacein space∆∆xx==tt∆∆vv≥≥ htht/(4/(4π π ∆∆xx00))

•• Note that Note that ∆∆xx is inversely is inversely proportional to proportional to ∆∆xx00

•• It means the more we know It means the more we know about the protonabout the proton’’s position s position at at tt = 0 the less we know = 0 the less we know about its later position at about its later position at t t > > 0. 0.

•• The original wave group has The original wave group has spread out to a much wider spread out to a much wider one because the phase one because the phase velocities of the component velocities of the component wave vary with wave wave vary with wave number and a large range of number and a large range of wave numbers must have wave numbers must have been present to produce the been present to produce the narrow original wave groupnarrow original wave group

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Broadening of spectral lines due to Broadening of spectral lines due to uncertainty principleuncertainty principle

•• An excited atom gives up it excess energy by emitting a An excited atom gives up it excess energy by emitting a photon of characteristic frequency. The average period that photon of characteristic frequency. The average period that elapses between the excitation of an atom and the time is elapses between the excitation of an atom and the time is radiates is 1.0radiates is 1.0××1010--88 s. Find the inherent uncertainty in the s. Find the inherent uncertainty in the frequency of the photon. frequency of the photon.

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SolutionSolution

•• The photon energy is uncertain by the amountThe photon energy is uncertain by the amount•• ∆∆EE ≥≥ hchc/(4/(4ccπ∆π∆tt)= 5.3)= 5.3××1010--27 27 J = 3J = 3.3.3××1010--8 8 eVeV•• The corresponding uncertainty in the frequency of light is The corresponding uncertainty in the frequency of light is

∆ν∆ν = = ∆∆EE//h h ≥≥ 88××10106 6 HzHz•• This is the irreducible limit to the accuracy with which we This is the irreducible limit to the accuracy with which we

can determine the frequency of the radiation emitted by an can determine the frequency of the radiation emitted by an atom. atom.

•• As a result, the radiation from a group of excited atoms As a result, the radiation from a group of excited atoms does not appear with the precise frequency does not appear with the precise frequency νν. .

•• For a photon whose frequency is, say, 5.0For a photon whose frequency is, say, 5.0××101014 14 Hz, Hz, •• ∆ν∆ν//νν =1.6=1.6××1010--88

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ExampleExampleEstimating quantum effect of a macroscopic particle

•• Estimate the minimum uncertainty velocity of a Estimate the minimum uncertainty velocity of a billardbillard ball (ball (mm ~ ~ 100 g) confined to a 100 g) confined to a billardbillard table of dimension 1 mtable of dimension 1 m

SolutionSolutionForFor ∆∆xx ~ 1 m, we have~ 1 m, we have∆∆pp ≥≥hh/4/4π∆π∆xx = 5.3x10= 5.3x10--3535 Ns,Ns,

•• SoSo ∆∆vv =(=(∆∆pp)/)/mm ≥≥ 5.3x105.3x10--3434 m/sm/s•• One can considerOne can consider ∆∆vv = 5.3= 5.3xx1010--3434 m/sm/s (extremely tiny) is the (extremely tiny) is the

speed of the speed of the billardbillard ball at anytime caused by quantum effectsball at anytime caused by quantum effects•• In quantum theory, no particle is absolutely at rest due to the In quantum theory, no particle is absolutely at rest due to the

Uncertainty PrincipleUncertainty Principle

1 m long billardtable

A billard ball of 100 g, size ~ 2 cm

∆∆vv == 5.3 5.3 x x 1010--3434 m/sm/s

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A particle contained within a finite A particle contained within a finite region must has some minimal KEregion must has some minimal KE

•• One of the most dramatic consequence of the One of the most dramatic consequence of the uncertainty principle is that a particle confined in uncertainty principle is that a particle confined in a small region of finite width cannot be exactly at a small region of finite width cannot be exactly at rest (as already seen in the previous example)rest (as already seen in the previous example)

•• Why? BecauseWhy? Because……

•• ...if it were, its momentum would be precisely ...if it were, its momentum would be precisely zero, (meaning zero, (meaning ∆∆pp = 0) which would in turn = 0) which would in turn violate the uncertainty principleviolate the uncertainty principle

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What is the What is the KKaveave of a particle in a box of a particle in a box due to Uncertainty Principle?due to Uncertainty Principle?

•• We can estimate the minimal KE of a particle confined in a box We can estimate the minimal KE of a particle confined in a box of size of size aa by making use of the U.P. by making use of the U.P.

•• If a particle is confined to a box, its location is uncertain byIf a particle is confined to a box, its location is uncertain by∆∆xx == a a

•• Uncertainty principle requires that Uncertainty principle requires that ∆∆pp≥≥ ((hh/2/2π)π)a a •• (don(don’’t worry about the factor 2 in the uncertainty t worry about the factor 2 in the uncertainty

relation since we only perform an estimationrelation since we only perform an estimation))

a

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ZeroZero--point energypoint energy( )

2

2

~

2

~

2

ave 222 mamp

mpK

av

>∆>⎟⎟⎠

⎞⎜⎜⎝

⎛=

This is the zero-point energy, the minimal possible kinetic energy for a quantum particle confined in a region of width a

Particle in a box of size a can never be at rest (e.g. has zero K.E) but has a minimal KE Kave (its zero-point energy)

We will formally re-derived this result again when solving for the Schrodingerequation of this system (see later).

a

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PYQ 2.11 Final Exam 2003/04PYQ 2.11 Final Exam 2003/04

• Assume that the uncertainty in the position of a particle is equal to its de Broglie wavelength. What is the minimal uncertainty in its velocity, vx?

• A. vx/4p B. vx/2p C. vx/8p• D. vx E. vx/p

• ANS: A, Schaum’s 3000 solved problems, Q38.66, pg. 718

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RecapRecap

•• Measurement necessarily involves interactions between Measurement necessarily involves interactions between observer and the observed systemobserver and the observed system

•• Matter and radiation are the entities available to us for such Matter and radiation are the entities available to us for such measurementsmeasurements

•• The relations The relations pp = = hh//λλ and and EE = = hhνν are applicable to both are applicable to both matter and to radiation because of the intrinsic nature of matter and to radiation because of the intrinsic nature of wavewave--particle dualityparticle duality

•• When combining these relations with the universal waves When combining these relations with the universal waves properties, we obtain the Heisenberg uncertainty relationsproperties, we obtain the Heisenberg uncertainty relations

•• In other words, the uncertainty principle is a necessary In other words, the uncertainty principle is a necessary consequence of particleconsequence of particle--wave dualitywave duality

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